The background description provided herein is for the purpose of generally presenting the context of the disclosure. Work of the inventors hereof, to the extent the work is described in the background section, as well as aspects of the description that may not otherwise qualify as prior art at the time of filing, are neither expressly or impliedly admitted as prior art against the present disclosure.
The concept of a digital communications channel is well known. In particular, it is known that a channel can affect the amplitude and phase of a signal carried by the channel. As a simple example, suppose a signal cos(ω0t′) is communicated in a channel, where w0 is the angular frequency of the signal, t′ is the time associated with the transmission of the signal, and t′=0 represents the beginning of the transmission. Ideally, the signal that arrives at a receiver should have the same amplitude and phase; i.e., the received signal should be cos(ω0t), where t is the time associated with receiving the signal, and t=0 represents the beginning of the reception. The time t=0 may correspond to t′=τ, for some transmission delay τ. However, the received signal is seldom the same as the transmitted signal, even in a noiseless environment. Rather, (in the absence of noise) a receiver will more likely receive a signal A cos(ω0t+ϕ), where A is a real number that shows the channel's effect on the amplitude of the signal, and ϕ is a real number that shows the channel's effect on the phase of the signal. The quantity ϕ is commonly referred to as “initial phase.”
Although the example above shows a transmission signal that has only one frequency component ω=ω0, a signal may include more than one frequency component. Additionally, a channel may affect each frequency component differently. Accordingly, the amplitude A and the initial phase ϕ in the example above may only apply to frequency component ω=ω0. From this point on, when a signal includes more than one frequency component, the amplitude and initial phase for each frequency component ω=ωi will be denoted with a corresponding subscript i.
A fundamental concept of digital communications is that amplitude and initial phase can be represented by a coordinate in a Cartesian plane. For example, an amplitude A and an initial phase ϕ can be represented by the coordinate (x,y) where x=A cos(ϕ) and y=A sin(ϕ). Conversely, given a coordinate (x,y), an amplitude and initial phase can be computed by A=√{square root over (x2+y2)} and
  ϕ  =            arctan      ⁡              (                  y          x                )              .  Another fundamental concept is that a coordinate (x,y) can also correspond to a complex number of the form (x+jy), where j is the imaginary unit. In this case, the x-axis represents the real part of a complex number, and the y-axis represents the imaginary part of a complex number. The benefits of representing amplitude and initial phase graphically as a coordinate point and mathematically as a complex number are that these representations allow changes in amplitude and initial phase to be easily illustrated and computed. The next paragraph shows an example of computing a channel's effects on a signal's amplitude and initial phase. In particular, an important computation involves Euler's formula, which states that a complex number (x+jy) can equivalently be expressed as Aejϕ, where, as shown above, A=√{square root over (x2+y2)} and
  ϕ  =            arctan      ⁡              (                  y          x                )              .  
As an example, suppose a transmitted signal in a channel has frequency components of the form Ai cos(ωit+ϕi). In the absence of noise, the channel will generally alter the amplitude multiplicatively by a factor Ki, and alter the initial phase additively by a factor θi, resulting in a received frequency component of the form KiAi cos(ωit+ϕi+θi). Representing these amplitudes and initial phases mathematically, the amplitude and initial phase of the transmitted frequency components can be characterized by Aiejϕ, and those of the received frequency component can be characterized by KiAiejϕi+θi)AiejϕiKiejθi. This shows two important things. First, it can be seen that the channel's effect on the amplitude and initial phase of the transmitted frequency component is captured by the term Kiejθi. Second, if (in the absence of noise) the amplitude and initial phase of a received frequency component is Biejφi, then the channel's effect on the transmitted amplitude and initial phase can be computed by
                              K          i                ⁢                  e                      j            ⁢                                                  ⁢                          θ              i                                          =                                                  B              i                        ⁢                          e                              j                ⁢                                                                  ⁢                                  φ                  i                                                                                        A              i                        ⁢                          e                              j                ⁢                                                                  ⁢                                  ϕ                  i                                                                    =                                            B              i                                      A              i                                ⁢                      e                          j              ⁡                              (                                                      φ                    i                                    -                                      ϕ                    i                                                  )                                                          ;          i      .      e      .        ,            K      i        =                                        B            i                                A            i                          ⁢                                  ⁢        and        ⁢                                  ⁢                  θ          i                    =                        φ          i                -                              ϕ            i                    .                    When all of the effects Kiejθi across a continuous frequency range are quantified, the result is a function showing a channel's effect on signal amplitude and initial phase based on frequency. The function is referred to in the art as a “transfer function.” A graph of a transfer function with respect to frequency is referred to as the channel's “frequency response.”
The examples above assume an absence of noise in or affecting the channel. As mentioned above, a signal's frequency component can have amplitude and initial phase that are represented by a complex number si, and the channel's frequency response for the frequency component can be represented by a complex number hi. In the absence of noise, the received frequency component has amplitude and initial phase given by yi=hi·si. However, a channel's frequency response can vary over time. Therefore, the value of hi may need to be re-evaluated. One way in which this can be accomplished is by sending the receiver a “training signal,” which is a predetermined signal that is known by the receiver. The training signal can include a frequency component that has predetermined amplitude and initial phase given by si. When the training signal arrives at the receiver with amplitude and initial phase yi, the receiver can evaluate the channel's frequency response for the frequency component by computing
      h    i    =                    y        i                    s        i              .  
However, when noise is present, the channel's frequency response becomes more difficult to estimate. As used herein, the term “noise” refers to phenomena or effects, in or affecting a channel, that affect a signal carried on the channel and that are not already included by the channel's frequency response. Generally, when noise is present, the received frequency component becomes yi=hi·si+ni. In this situation, both hi and ni may vary over time, and it becomes more difficult to estimate the frequency response hi with certainty based on knowing only the training component si and the received component yi. In some embodiments, the noise ni may be represented by the symbol zi. In some embodiments, the noise may also introduce a large and possibly random phase difference into the received signal yi.
Ultimately, the desired operation of a receiver is to correctly detect a transmitted signal. To do so, a receiver can benefit from having a more accurate estimate of the channel's frequency response. Additionally, a channel estimate is useful for many kinds of operations, such as equalization. However, the presence of noise undermines the receiver's ability to produce an accurate channel estimate. Accordingly, there is continued interest in improving a receiver's channel estimation capabilities.